Systems integrating several sensors are used in a great variety of sectors such as site surveillance, maintenance, robotics, medical diagnosis or meteorological forecasts. Such systems carry out, for example, functions of classification, identification and tracking in real time.
To best exploit multi-sensor systems, it is necessary to use an effective scheme for fusion of information so as to combine the data originating from the various sensors of the system and generate a decision.
According to the known art, certain information fusion schemes rely on Dempster-Shafer theory and thus use belief functions, which is the basic tool for representing information in this theory. Belief functions are known for their capacity to faithfully represent imperfect information. Information comprising an imprecision or an uncertainty or incomplete information is called imperfect information. The sensors of a system are considered to be imperfect information sources, notably because of their imprecision. The term sensor is understood here in the broad sense. It includes physical devices for data acquisition (camera, radar, etc.) but also devices for processing these data. It is possible to establish a belief function on the basis of the data provided by most commercially available sensors. Schemes for combining belief functions may be used. By dint of their nature, these schemes are therefore particularly appropriate to the problem of the fusion of imperfect information arising from sensors.
Let X be a variable with values in a finite set Ω. The information held by a sensor as regards the value actually taken by X may be quantified by a belief function. A belief function is formally defined as a function, denoted bel, from the power set of Ω, denoted 2Ω, in the interval [0,1] and satisfying certain mathematical properties. The quantity bel(A) represents the total degree of belief allocated by the sensor to the fact that the value of X is in A, where A is a part (also called a subset) of Ω (this being written A⊂Ω). There exist various equivalent representations of a belief function, which are useful in practice. In particular, the mass function, denoted m, is a function from 2Ω to [0,1] which satisfies the condition:
            ∑              A        ⊆        Ω                                  ⁢                  ⁢          m      ⁡              (        A        )              =  1.
The quantity m(A), called the mass of A, represents the degree of belief allocated to A (and to no strict subset). The belief function bel associated with a mass function m is obtained in the following manner:bel(A)=ΣØ≠B⊂Am(B),for all A⊂Ω. The mass function m associated with a belief function bel is obtained in the following manner:m(A)=ΣØ≠B⊂A(−1)|A|−|B|bel(B),for all A⊂Ω. By dint of the one-to-one correspondence between a belief function bel and its associated mass function m, we will use the name Belief Function (BF) in the broad sense in this document to designate both a belief function bel and also its associated mass function m (the context is generally sufficient to determine whether the objects manipulated are mass functions m or belief functions bel).
It is possible to consider by way of nonlimiting example, a multi-sensor system of classifier type used for the optical recognition of hand-written characters. It is assumed that the system is intended to determine whether the character formed on an image I (not represented) is one of the letters ‘a’ or ‘b’. Let X be the variable associated with the character. We therefore have a set of values Ω={a, b} for the variable (the character) X. Each of the sensors of the system is a classifier which itself provides an item of information regarding the character to be recognized in the form of a BF. It is assumed that there are two sensors of this type in our example. The sensor C1 provides a belief function mC1 defined as follows:mC1(Ø)=0mC1({a})=0.8mC1({b})=0mC1({a,b})=0.2
In this example, the sensor C1 considers that it is highly probable (0.8) that the character to be recognized is {a}. mC1({a,b})=mC1(Ω)=0.2 represents the sensor's ignorance. The sensor C2 provides a belief function mC2 defined as follows:mC2(Ø)=0mC2({a})=0.2mC2({b})=0mC2({a,b})=0.8
The function mC2 conveys the fact that the sensor C2 has fairly little information regarding the character observed: the sensor ascribes a small degree of belief (0.2) to the character {a} and a large degree of belief (0.8) to ignorance, that is to say to the set Ω={a, b}.
Dempster-Shafer theory makes it possible to combine the belief functions representing information arising from different sources, so as to obtain a belief function taking into account the influences of each of the sources. The belief function thus obtained, called the merged belief function, represents the combined knowledge of the various imperfect information sources (the sensors).
In order to obtain a merged belief function representative of reality, it is necessary to take into account knowledge pertaining to the states of the sources.
The known information fusion schemes only make it possible to take into account certain types of knowledge regarding the states of the sources (sensors). Indeed, the existing solutions are limited to the consideration of particular knowledge regarding the dependency, the competence and the sincerity of the sources. For example, a merge operator making it possible to merge information arising from independent and competent sources is already known. This operator is known in the literature by the name “unnormalized Dempster's rule”. We denote this operator by the symbol ⊕D. Thus, the BF resulting from the fusion by ⊕D of two BFs mC1 and mC2 is denoted mC1⊕DmC2. The definition of the operator ⊕D is as follows. Consider two belief functions mC1 and mC2, we have, for all A⊂Ω:
            m              C        ⁢                                  ⁢        1              ⁢          ⊕      D        ⁢                  m                  C          ⁢                                          ⁢          2                    ⁡              (        A        )              =            ∑                        B          ⋂          C                =        A                                  ⁢                  ⁢                            m                      C            ⁢                                                  ⁢            1                          ⁡                  (          B          )                    ⁢                                    m                          C              ⁢                                                          ⁢              2                                ⁡                      (            C            )                          .            
Applying the formula to the two belief functions mC1 and mC2 given as an example above, we obtain:
                                                        m                              C                ⁢                                                                  ⁢                1                                      ⁢                          ⊕              D                        ⁢                                          m                                  C                  ⁢                                                                          ⁢                  2                                            ⁡                              (                                  {                  a                  }                                )                                              =                    ⁢                                                                      m                                      C                    ⁢                                                                                  ⁢                    1                                                  ⁡                                  (                                      {                    a                    }                                    )                                            ⁢                                                m                                      C                    ⁢                                                                                  ⁢                    2                                                  ⁡                                  (                                      {                    a                    }                                    )                                                      +                                                            m                                      C                    ⁢                                                                                  ⁢                    1                                                  ⁡                                  (                                      {                    a                    }                                    )                                            ⁢                                                m                                      C                    ⁢                                                                                  ⁢                    2                                                  ⁡                                  (                                      {                                          a                      ,                      b                                        }                                    )                                                      +                                                                  ⁢                                                    m                                  C                  ⁢                                                                          ⁢                  1                                            ⁡                              (                                  {                                      a                    ,                    b                                    }                                )                                      ⁢                                          m                                  C                  ⁢                                                                          ⁢                  2                                            ⁡                              (                                  {                  a                  }                                )                                                                                  =                    ⁢                                    0.8              *              0.2                        +                          0.8              *              0.8                        +                          0.2              *              0.2                                                                    =                    ⁢          0.84                                                                            m                              C                ⁢                                                                  ⁢                1                                      ⁢                          ⊕              D                        ⁢                                          m                                  C                  ⁢                                                                          ⁢                  2                                            ⁡                              (                                  {                                      a                    ,                    b                                    }                                )                                              =                                                    m                                  C                  ⁢                                                                          ⁢                  1                                            ⁡                              (                                  {                                      a                    ,                    b                                    }                                )                                      ⁢                                          m                C2                            ⁡                              (                                  {                                      a                    ,                    b                                    }                                )                                                                                  =                      0.8            *            0.2                                                        =          0.16                    
We also deduce mC1⊕DmC2({b})=0 and mC1⊕DmC2(Ø)=0 through the condition
            ∑              A        ⊆        Ω                                  ⁢                  ⁢          m      ⁡              (        A        )              =  1for every BF m.
A merge operator making it possible to merge information arising from independent sources at least one of which is competent is also already known. This operator is known in the literature by the name “disjunctive rule”. We denote this operator by the symbol ⊕DP.
A more general operator than the operators ⊕D and ⊕DP is also already known, making it possible to take into consideration knowledge such as the propensity of the sensors to be in some such state in regard to their competence and their sincerity, rather than in some such other state. This operator, however, requires that the sensors be independent. For example, if we denote by E1 the state “the sources are competent and independent” and by E2 the state “the sources are independent and at least one is competent”, this operator can take into account knowledge of the type: “with a probability p, the sources are in the state E1 (i.e. with a probability p, the sources are competent and independent), and with a probability 1−p, they are in the state E2 (i.e. with a probability 1−p the sources are independent and at least one is competent)”.
A merge operator making it possible to merge information arising from competent and non-independent sources is also already known. This operator is known in the literature by the name “cautious rule”. We denote this operator by the symbol ⊕P. Thus, the BF resulting from the fusion by ⊕P of two BFs mC1 and mC2 is denoted mC1⊕PmC2. The definition of the operator ⊕P being complex in the general case, but simple in the case of belief functions of the type of those given as an example above, we are content here merely to give the definition of the cautious rule for belief functions of the type of those given as an example above. Consider two belief functions mC1 and mC2 on Ω={a, b} such that:mC1({a})=1−y, mC1({a,b})=y andmC2({a})=1−z, mC2({a,b})=z, with y,zε(0,1) (we therefore have mC1(Ø)=0, mC1({b})=0, and mC2(Ø)=0, mC2({b})=0 through the condition
            ∑              A        ⊆        Ω                                  ⁢                  ⁢          m      ⁡              (        A        )              =  1for every BF m). The result mC1⊕PmC2 of the fusion by the cautious rule of the BFs mC1 and mC2 is then obtained with the formula:mC1⊕PmC2({a})=1−minimum(y,z),mC1⊕PmC2({a,b})=minimum(y,z).
Applying the formula to the two belief functions mC1 and mC2 given as an example above, we obtain:mC1⊕PmC2({a})=1−minimum(0.8,0.2)=1−0.2=0.8,mC1⊕PmC2({a,b})=minimum(0.8,0.2)=0.2.Note that there also already exists an operator, called the “bold rule”, making it possible to merge information arising from non-independent sources at least one of which is competent.
However, these known fusion schemes do not make it possible to take into account all the types of knowledge regarding the dependency, the competence and the sincerity of the sources. For example, if we denote by E1 the state “the sources are competent and non-independent” and by E2 the state “the sources are independent and at least one is competent”, the known schemes do not make it possible to process the knowledge: “with a probability p, the sources are in the state E1 and with a probability 1−p, they are in the state E2”, although the operators which correspond to the states E1 and E2 (respectively, the cautious rule and the disjunctive rule) are already known. More generally, it is possible to define E={E1, . . . , EN} the set of possible operating states of the sensors and ⊕Ei the operator corresponding to the state Ei where the states Ei, i=1, . . . N, considered do not necessarily deal with the dependency, the competence and the sincerity of the sources. In this case, no general scheme exists making it possible to merge the imperfect information arising from the sensors when we have knowledge regarding the states of the sensors of the type “with a probability pi, the sensors are in the state Ei, i=1, . . . N”.